If you plot the strong force coupling constant's strength against energy scale, near its peak it looks like a bell curve (with a linear strength scale and logarithmic energy scale), with a peak close to 1 (maybe even as much as 1.25) at roughly q

^{2}=0.5 GeV

^{2}, that declines in a very long tail towards higher energies, and in a much shorter tail towards lower energies. It isn't entirely resolved whether the strong force coupling constant takes a value that is 0 (a value sometimes called "trivial") or some fixed value greater than zero in the limit at an energy scale of q

^{2}=0 (called an "infrared fixed point"). The value that the strong force coupling constant takes at the Z boson mass of 91.1876(21) GeV on the ultraviolet side of the peak is roughly the same as the value it takes at a hair over q

^{2}=0.01 GeV

^{2}on the infrared side of the peak.

The most accurate world average measurement of this quantity (as of 2014) at an energy scale equal to the Z boson mass is 0.1185 +/- 0.0006. By comparison, twenty years ago, back in 1994, the most accurate world average measurement was 0.123 +/- 0.006. So, since then, the best fit value has fallen by about 3.8% and the precision of the measurement has improved by a factor of ten (the values are consistent with each other to within the margin of error). We've improved, but only a little. At this rate, we'll have a measurement as accurate as the current electromagnetic coupling constant measurement in another ten million years, and will reach the accuracy with which we have measured the Newton's constant of gravitation in about 250 years.

This is about 20 times less precise than the most accurate combined measurement of the W boson mass. It about 50 times less precise than the most accurate combined measurement of the Newton's constant in general relativity. It is about 1,200,000 times less precise than the the most accurate combined measurement coupling constant for electromagnetism, and it is about 100,000 times less precise than the most accurate combined measurement of the coupling constant for the weak force.

The only constants in the Standard Model known less precisely than the strong force coupling constant are some of the quark masses, some of the tiniest CKM matrix entries, and some of the neutrino physics constants.

The latest ATLAS measurement of this quantity as 7 TeV is 0.1173 +/- 0.0046, with a theoretical error of about four times as much as the experimental error. This is consistent with previous measurements at the one sigma (i.e. one standard deviation) level.

This ATLAS experiment in question was primarily measuring something else and wasn't designed to produce a world class strong force coupling constant measurement, so the 4% margin of error doesn't reflect badly on ATLAS and confirms once again that Standard Model constants measured in many different ways have the same values within error bars, confirming that it is robust and that the quantity defined in theory corresponds to a meaningful quantity in the real world.

The great inaccuracy with which the strong force coupling constant is known does, however, does represent a fundamental barrier to accurate QCD calculations. For example, the proton mass can be calculated from QCD first principles to a precision of only about 1%, about half of which is attributable to uncertainty in the strong force coupling constant, even though the proton mass has been directly measured to a precision of eight significant digits.

And, while this would seem like it is the fault of imprecise experiments, it really isn't. We have measured observable properties of all manner of hadrons (particle made of quarks that are bound together by the strong force such as protons, neutrons and pions), which are much more precise. But, there is so much theoretical uncertainty in how to correctly calculate those measured observables that it isn't possible to reverse engineer them into very precise measurements of quark masses and the strong force coupling constant.

This isn't for want of trying. Theoretical physicists like the author of the 4gravitons blog devote the lion's share of their professional efforts every day to "come up with mathematical tricks to make particle physics calculations easier." Physicist have been working steadily on this problem for half a century and have made only modest progress, although steady advances in computers computational power over that time period have helped a great deal. But, given that computation power has increasing according to Moore's law by a factor of about 2.6 billion over that time period since QCD was invented, progress has still been painstakingly slow. The math really is just that hard.

**The Strong Coupling Constant and BSM Theories**

You might think that people crafting beyond the Standard Model physics theories would be swarming to produce theories that would predict slightly higher or slightly lower strong force coupling constants because this is an area where there is considerable room to make contradictory predictions within the existing margin of error. But, in fact, there is almost no activity on this front and beyond the Standard Model physics proposals tend to leave the strong force coupling constant and the QCD equations of the strong force, more or less unchanged apart from the number of strongly interacting particles. There isn't even a great deal of work being done that makes differing predictions regarding the quark masses.

Supersymmetry theories, generically, predict that the running of the strong force coupling constant will take place at a different rate (i.e. that it will have a different "beta function") than in the Standard Model, which may be possible to compare to the Standard Model prediction at LHC Run II. Basically, the strong force coupling constant gets weaker with higher energies in the Minimal Supersymmetric Model (MSSM) at about half the rate that it does in the Standard Model (the weak force coupling constant gets weaker at higher energies in the Standard Model but stronger at higher energies in the MSSM, and the electromagnetic force coupling constant which gets stronger at higher energies in both theories gets 30% stronger at given higher energies in the MSSM than in the Standard Model).

The actual value of the strong force coupling constant, however, is actually harder to measure at high energies than at low energies, because using the Standard Model beta function for the strong force coupling constant, a comparatively wide range of low energy values at lower energies when run according to the strong force coupling constant beta function of the Standard Model, all converge at points which are extremely close to each other at much high energies. So high energy measurements of the strong force coupling constant must be much more precise at higher energies than at lower energies to make a measurement of the same precision. As a result, LHC Run II probably won't do much to improve the accuracy with which the strong force coupling constant is measured.

Many beyond the Standard Model physics proposals either don't answer these questions at all, leaving them as experimentally measured physical constants, just as the Standard Model does, or assumes relationships between these valuable that are right at an order of magnitude level but impossible to calculate more precisely at this time. Phenomenologists try to look for patterns in the currently measured values of these constants, but even there, the results have thus far been underwhelming.

The three coupling constants in superysmmetric theories have one less degree of freedom than in the Standard Model (i.e. two instead of three). This is because, in principle, in supersymmetric theories, the strong force coupling constant can be calculated exactly from an ultraviolet fixed point where the strength of the strong, weak and electromagnetic force coupling constants are the same at the "GUT" scale of about 10

^{16}GeV, using the supersymmetric strong force coupling constant beta function. The coupling constant strength and the fixed point energy scale could be calculated using supersymmetric beta functions for the weak force coupling constant and electromagnetic force coupling constant whose values are known much more precisely than the strong force coupling constant. This fixed point, calculated from much more precise experimental inputs, could in turn be used to back out a strong force coupling constant at a measurable energy which should be much more precise than the experimentally measured version. The result should be almost 10,000 times as precise as the current experimentally measured values, since the beta functions, in both the Standard Model and supersymmetry, in principle, ought to be possible to determine exactly without experimental inputs from the theory.

I have not seen a paper making the calculation this way, although it should be straightforward to do once the supersymmetric beta functions are established. This calculation wouldn't be useful from a predictive perspective, because there are so many different variations on the supersymmetry theme out there. But, it would make it possible to use the measured strong force coupling constant value at low energy scales to discriminate experimentally between the possibilities, thus narrowing "theory space" and the parameter space of theories. Perhaps existing supersymmetric parameter fitting software does this in a way that isn't transparent to a casual observer like me.

My own conjecture (meaning that I am someone who thinks that this might be true, not that I am the first or only person to have come up with it) is that the three Standard Model force coupling constants do actually converge at a GUT scale, but that this happens only once quantum gravity effects on the ultraviolet running of these constants are considered, which the Standard Model beta functions do not.

Newton's constant is generally a running coupling constant in quantum gravity theories, something which greatly impacts the physics of the very early universe, and a number of beta functions, such as this one, have been proposed.

## 15 comments:

"the three Standard Model force coupling constants do actually converge at a GUT scale, but that this happens only once quantum gravity effects on the ultraviolet running of these constants are considered, which the Standard Model beta functions do not."

Seriously?

Why would there be any quantum gravity effects on the ultraviolet running of those constants?

Quantum gravity (or, for that matter, any new force or particle) automatically adds a new term into the renormalization group formula applies to the Standard Model Lagrangian which is used to compute the beta function for each of the other three forces.

Also, at the GUT scale, the energy density of the interacting particles becomes so great that gravitational effects can't be ignored.

And, keep in mind that the amount of tweak required isn't huge. You are talking about adjustments on the order of 1% over the entire 16 order of magnitude range for one of the beta functions, or less spread over all three, to make this happen.

In other news, a precision extra-solar test of the gravitational constant from a white dwarf-pulsar system 3,750 light years away, confirms that Newton's constant, G, is the same there as it is here, although the press release does not make clear the precision of the extra-solar measurement.

The pre-print from last April is here.

A good survey of the physics of how quantum gravity could impact the running of other coupling constants can be found here. A Wikipedia article notes the existence of a quantum gravity correction to the running of the fine structure constant in the asymptotically safe gravity approach to quantum gravity.

I don't follow your scheme for getting highly precise predictions for alpha_s from a GUT, susy or otherwise... A supersymmetric GUT, in particular, will have many free parameters. The measured coupling constants are constraints on the possible values of those parameters, as is the absence of proton decay, as are various other measurements, and all these things taken together can even rule out certain GUT possibilities, as there is nowhere in the GUT parameter space consistent with observed reality. But I simply don't see a way to do what you describe.

See this exposition of how it works in the simplest case, non-susy SU(5). The GUT has one fundamental coupling constant, g5. The SM couplings at the energy scale where GUT symmetry breaking occurs can all be written in terms of g5, and at lower energies they run as in the SM. But this GUT breaking scale itself depends on several free parameters, appearing in the interaction Lagrangian for the superheavy Higgs that does the breaking. This simplest case doesn't work anyway, since the SM couplings simply don't meet, though they come close. Therefore people have considered susy, or non-susy SO(10) with an intermediate scale of Pati-Salam partial unification, etc.

"I don't follow your scheme for getting highly precise predictions for alpha_s from a GUT, susy or otherwise."

You need the following:

1. The experimental world average of the electromagnetic coupling constant at a given energy scale.

2. The experimental world average of the weak force coupling constant at a give energy scale.

3. The exact beta function of the electromagnetic coupling constant.

4. The exact beta function of the weak force coupling constant.

5. The exact beta function of the strong force coupling constant.

6. Knowledge that the theory has gauge coupling unification at a single point from theoretical considerations, which is generically true in SUSY theories, but is not true in many other classes of theories. This procedure does not work in theories that do not include as a feature triple gauge coupling constant unification. Many non-SUSY GUT theories such as the one referenced in your comment do not have this feature and hence have three degrees of coupling constant freedom rather than two.

7. Key observation. The beta function of the coupling constants of a theory (such as the SM or MSSM or any other particular SUSY or GUT theory) can in principle be determined exactly without reference to the experimentally measured parameters of that theory. The parameters of the beta functions of these theories are not experimentally measured parameters of the theory.

Process. Use 1-4, to determine the exact energy scale at which the electromagnetic force coupling constant and weak force coupling constant are identical (i.e. calculate the GUT scale of the theory), and the value of these coupling constants that that energy scale. As a consequence of 6, you can use 5 to reverse engineer the value of the strong force coupling constant at any value given knowledge of the energy scale and coupling constant value determined with 1-4.

Result. The strong force coupling constant can be determined at any arbitrary energy scale with an uncertainty only equal to the uncertainty in the least accurately known of 1 and 2 (currently the weak force coupling constant which is known with precision roughly 100,000 times that of the strong force coupling constant).

This sets out my reasoning which seems pretty much irrefutable if all of my assumptions are true. My assumption is that you doubt the veracity of 6 or 7, or at least, did not recognize that this process is limited to cases when 6 and 7 are true.

Also, to be clear, I am not claiming that any particular SUSY theory or GUT theory is actually true. I am proposing an experimental test of the truth of any given theory of that type that meets assumptions 6 and 7. If 6 and 7 is true, and the predicted value of the strong force coupling constant differs materially from the measured value of the strong force coupling constant, then that theory is ruled out.

In concrete terms, a SUSY theory is ruled out with 95% confidence if the predicted strong force coupling constant value from the method above is not between 0.1173-0.1197 and is ruled out with 99% confidence if the predicted strong force coupling constant value is not between 0.1167-0.1203.

This means that in the universe of SUSY theories (and more generally triple gauge coupling unified theories) for which exact beta functions for the coupling constants are known, each could be tested against experiment and either found to be consistent with the data or ruled out, in something on the order of an hour or less, without further analysis. My sense is that this has not been done comprehensively, and that it would rule out a large share of classes of potential SUSY theories.

For example, I know with some confidence that the exact beta functions of some of the most common SUSY theories (MSSM, NMSSM, CNMSSN) are known and therefore ought to be subject to quick and easy experimental testing by this means. I personally don't know if they would pass the test or not. I presume that MSSM would pass this test, but I am less clear about the others.

Andrew, I think that in fact your procedure *is* valid for non-susy SU(5), although no-one actually reasoned that way. Instead, they just employed the fact that below the unification point, the beta functions are those of the SM, extrapolated all three couplings to high energies, and showed that they don't meet, as they should in SU(5). But certainly what you suggest could be done - only extrapolate electromagnetic and weak upwards, locate the unification scale, then extrapolate downwards to what the strong coupling should be.

Basically, gauge unification is implied by GUT, not by Susy. At the level of theory, MSSM doesn't imply gauge unification any more than SM does. What *is* true is that the measured couplings, evolving under SM beta functions, do not converge exactly, whereas they can do so in the MSSM, because of the influence of the extra particle species on the beta functions. But to explain this convenience at the level of theory, you still need a GUT. It's just that, having started with MSSM, it must be a GUT with susy added.

Incidentally, because susy hasn't appeared, there is a small revival of non-susy GUTs happening. But this doesn't include original SU(5), it only includes more complex ones like SO(10).

*convergence, not convenience

I suppose you have a point. Gauge unification is almost definitionally part of a GUT.

On the other hand, if there are quantum gravity corrections to the beta functions, the "right" GUT without quantum gravity corrections, would not necessarily display gauge unification. Gauge unification in a GUT that is not a TOE, might be "too good to be true" evidence that actually hurts that particular theory.

A new paper applies LHC data to gauge unification in the MSSM. It finds that the "SUSY Scale" at which SUSY effects should appear is 2.3 to 3.5 TeV with a best fit of 2.8 TeV.

http://arxiv.org/pdf/1508.04176.pdf This would imply gauge unification at 10^16 GeV and a unified coupling constant value of 1/25.83. This SUSY scale is high enough to be consistent with the fact that no SUSY phenomena have been observed so far.

But, less notable than the minimum SUSY scale estimate, is the maximum. Strong force coupling constant running constrains SUSY scale from both the top and the bottom. The data in 1991 didn't strongly constrain this, with a range of 100 GeV to 10 TeV, but the LHC data provide a narrower range that combined with other SUSY searches could easily become over constrained, even with just LHC Run 2 data.

Run I of the LHC provided strong force coupling constant experimental data up to 1.4 TeV, and the threshold will get higher in Run 2.

For example, current electron dipole moment experiments naively constrain the SUSY scale to be about 10 TeV, which would be inconsistent with the SUSY scale range permitted based upon the running of the strong force coupling constant.

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